The Discontinuous Galerkin (DG) Method is increasingly used nowadays to solve advection-dominated problems. Its most attractive features, namely the high accuracy obtained without the use of an extended stencil and the inherent parallelization, have turned it into a method of choice for Computational Fluid Dynamics problems (see the recent review in [Cockburn,Karniadakis,Shu (2001)]). The early formulation of the DG method by Cockburn and Shu (1989) uses as degrees of freedom the coefficients of the solution in a polynomial basis expansion (Legendre polynomials being used in one dimension to take advantage of their orthogonality). This modal formulation seems more intuitive, but has a large drawback for nonlinear problems: to compute nonlinear fluxes, one needs to first compute the value of the solution by evaluating the polynomial expansion at the flux integration points, then evaluate the fluxes. Researchers realized later that this translation from modal space into physical space can be avoided through a collocation (nodal) formulation. In the latter framework the degrees of freedom are the values of the solution at the collocation points, so the flux values can be evaluated easily, in particular if the collocation points are chosen to coincide with the integration points [Stanescu,Hussaini,Farassat (2003)], [Kopriva,Woodruff,Hussaini (2001)].

In the context of multigrid methods, the modal DG formulation again is more intuitive: the polynomial-type basis functions can be made hierarchical, and a p-multigrid approach is natural (the restriction operator for example just neglects extra modes); see recent research in this direction [Helenbrook,Mavriplis,Atkins (2003)], [Fidkowski,Darmofal (2003)]. Due to the advantage of the nodal approach for nonlinear problems, we want to investigate the feasibility of a p-type multigrid method in this latter formulation. It seems that such an investigation has not been reported yet in the literature. The use of an acceleration technique such as multigrid may reduce the difference in cost between the two approaches; however, this is very likely to be problem dependent, hence arguably a nodal DG p-multigrid method may lead to savings in computer time.

For a preliminary numerical result, we solve the one-dimensional nonlinear Euler equations and compare the performance (CPU time, MG cycles) of a nodal DG p-multigrid with a modal DG p-multigrid. Figure 1 shows the accuracy versus the number of MG cycles. Figure 2 shows the accuracy versus CPU time.

Figure 1: nodal DG p-multigrid (top) vs. modal DG p-multigrid (bottom) in the MG cycles

Figure 2: nodal DG p-multigrid (top) vs. modal DG p-multigrid (bottom) in CPU time